We have problem with cheat as well. The only way we get rid of it was either plow it under or Glysophate.
I sprayed 40 acres early this spring that was getting a lot of cheat that is going to corn.
Just tilled the field for corn a couple of hours ago and all the cheat was dead.
I don"t know if you ever get rid of in brome without starting over. It is going to fight you every spring. It grows from seed also from the roots of cut off plants.
It is called Cheat for a reason. It cheats and comes up early and chokes out any later crops.